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1

WB JEE 2022

MCQ (More than One Correct Answer)
English
Bengali

Consider the equation $$y - {y_1} = m(x - {x_1})$$. If m and x1 are fixed and different lines are drawn for different values of y1, then

A
the lines will pass through a fixed point
B
there will be a set of parallel lines
C
all lines intersect the line x = x1
D
all lines will be parallel to the line y = x1

$$y - {y_1} = m(x - {x_1})$$ সমীকরণটি বিবেচনা কর। যদি m ও x1 অপরিবর্তনীয় হয় ও y1 এর বিভিন্ন মানের জন্য ভিন্ন ভিন্ন সরলরেখা অঙ্কিত করা হয় তবে

A
সরলরেখাগুলি একটি নির্দিষ্ট বিন্দু দিয়ে যাবে
B
সমান্তরাল সরলরেখাগুচ্ছের একটি সেট পাওয়া যাবে
C
x = x1 সরলরেখাকে সমস্ত সরলরেখাগুলি ছেদ করবে
D
সমস্ত সরলরেখাগুলি y = x1 এর সমান্তরাল হবে
2

WB JEE 2020

MCQ (More than One Correct Answer)
English
Bengali
The equation of the straight line passing through the point (4, 3) and making intercepts on the coordinate axes whose sum is $$ - 1$$ is
A
$${x \over 2} - {y \over 3} = 1$$
B
$${x \over { - 2}} + {y \over 1} = 1$$
C
$$ - {x \over 3} + {y \over 2} = 1$$
D
$${x \over 1} - {y \over 2} = 1$$

Explanation

Equation of straight line having intercepts on the coordinate axes are a and b, are

$${x \over a} + {y \over b} = 1$$ ...(i)

Given, $$a + b = - 1 \Rightarrow b = - (a + 1)$$

Line (i) passing through the point (4, 3).

$$ \therefore $$ $${4 \over a} + {3 \over b} = 1$$

$$ \Rightarrow {4 \over a} - {3 \over {a + 1}} = 1$$

$$ \Rightarrow 4(a + 1) - 3a = a(a + 1)$$

$$ \Rightarrow {a^2} + a - a = 4$$

$$ \Rightarrow a = \pm 2$$

When $$a = 2$$, then $$b = - 3$$

and when $$a = - 2$$, then $$b = 1$$

$$ \therefore $$ Equation of straight line are $${x \over 2} - {y \over 3} = 1,{x \over { - 2}} + {y \over 1} = 1$$

(4, 3) বিন্দুগামী একটি সরলরেখার স্থানাঙ্ক অক্ষদ্বয়ের ছেদিতাংশের সমষ্টি হল $$-$$1 । সরলরেখাটির সমীকরণ হবে

A
$${x \over 2} - {y \over 3} = 1$$
B
$${x \over { - 2}} + {y \over 1} = 1$$
C
$$ - {x \over 3} + {y \over 2} = 1$$
D
$${x \over 1} - {y \over 2} = 1$$

Explanation

সরলরেখার সমীকরণ $${x \over a} + {y \over b} = 1$$

$$\therefore$$ $$a + b = - 1$$ ...... (i)

এবং $${4 \over a} + {3 \over b} = 1$$ বা, $$4b + 3a = ab$$

বা, $$4b + 3( - 1 - b) = ( - 1 - b)b$$

বা, $${b^2} + 2b - 3 = 0 \Rightarrow b = - 3,1$$

এবং $$a = 2, - 2$$

3

WB JEE 2019

MCQ (More than One Correct Answer)
English
Bengali
Straight lines x $$-$$ y = 7 and x + 4y = 2 intersect at B. Points A and C are so chosen on these two lines such that AB = AC. The equation of line AC passing through (2, $$-$$7) is
A
x $$-$$ y $$-$$ 9 = 0
B
23x + 7y + 3 = 0
C
2x $$-$$ y $$-$$ 11 = 0
D
7x $$-$$ 6y $$-$$ 56 = 0

Explanation

Given equations of lines are

x $$-$$ y = 7 .... (i)

and x + 4y = 2 ...(ii)

By solving Eqs. (i) and (ii), we get the point B(6, $$-$$1)


Let the slope of AC be m

then, AB = AC

$$ \therefore $$ $$\left| {{{m + {1 \over 4}} \over {1 - {m \over 4}}}} \right| = \left| {{{ - {1 \over 4} - 1} \over {1 - {1 \over 4}}}} \right|$$

$$ \Rightarrow m = {{ - 23} \over 7},1$$

When $$m = {{ - 23} \over 7}$$, then equation of line

$$y + 7 = {{ - 23} \over 7}(x - 2)$$

$$ \Rightarrow 7y + 49 = - 23x + 46$$

$$ \Rightarrow 23x + 7y + 3 = 0$$

When m = 1, then equation of line

$$y + 7 = (x - 2)$$

$$ \Rightarrow x - y - 9 = 0$$

$$x - y = 7$$ ও $$x + 4y = 2$$ সরলরেখাদ্বয় B বিন্দুতে পরস্পরকে ছেদ করে। ওই দুই রেখার উপর A ও C বিন্দু দুটি এমনভাবে নেওয়া হল যে, $$AB = AC$$ হয়। $$(2, - 7)$$ বিন্দুগামী $$AC$$ রেখার সমীকরণ হল -

A
$$x - y - 9 = 0$$
B
$$23x + 7y + 3 = 0$$
C
$$2x - y - 11 = 0$$
D
$$7x - 6y - 56 = 0$$

Explanation

$$x - y = 7$$ ...... (i)

$$x + 4y = 2$$ .... (ii)

$$(2, - 7)$$ বিন্দুগামী $$AC$$ সরলরেখা সমীকরণ

ধরি, $$y + 7 = m(x - 2)$$ ..... (iii)

$$\Delta ABC$$ এর $$AC = AB$$ $$\therefore$$ $$\angle B = \angle C$$

(i) এর প্রবণতা = 1, (ii) এর প্রবণতা $$ = - {1 \over 4}$$, (iii) এর প্রবণতা $$ = m$$

$$\therefore$$ $$\left| {{{m + {1 \over 4}} \over {1 - {1 \over 4}m}}} \right| = \left| {{{1 + {1 \over 4}} \over {1 - {1 \over 4}}}} \right|$$

বা, $$\left| {{{4m + 1} \over {4 - m}}} \right| = \left| {{{{5 \over 4}} \over {{3 \over 4}}}} \right| = {5 \over 3}$$

$$\therefore$$ $${{4m + 1} \over {4 - m}} = \pm \,{5 \over 3}$$

বা, $$3(4m + 1) = \pm 5(4 - m)$$

বা, $$17m - 17 = 0$$ অথবা $$7m + 23 = 0$$

$$\therefore$$ $$m = 1$$ এবং $$ - {{23} \over 7}$$

$$\therefore$$ সরলরেখার সমীকরণ $$y + 7 = (x - 2)$$ অথবা $$y + 7 = - {{23} \over 7}(x - 2)$$ বা, $$x - y - 9 = 0$$ অথবা $$23x + 7y + 3 = 0$$

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