Two equal sides of an isosceles triangle are along – x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is: [2025]

Question

Two equal sides of an isosceles triangle are along – x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is: [2025]

Smart Achievers · Accepted Answer

(4)

Slope of given lines are $m_{1} = \frac{1}{2}$ and $m_{2} = - 1$ .

Since, AB = AC, then $∠$ ABC = $∠$ ACB

Now, angle between AB and BC = angle between AC and BC

$\frac{m - \frac{1}{2}}{1 + \frac{1}{2} (m)} = \frac{- 1 - m}{1 - m}$

$\Rightarrow \frac{2 m - 1}{2 + m} = \frac{m + 1}{m - 1}$

$\Rightarrow 2 m^{2} - 3 m + 1 = m^{2} + 3 m + 2$

$\Rightarrow m^{2} - 6 m - 1 = 0$

Sum of roots = 6

$∴$ Required sum is 6.

Solution

Q.
Two equal sides of an isosceles triangle are along – x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is: [2025]

1 $- 2 \sqrt{10}$

2 12

3 –6

4 6

Want Us to Email you About Special Offers & Updates?

Solution

Q. Two equal sides of an isosceles triangle are along – x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is: [2025]

1 –210

2 12

3 –6

4 6

Ans. (4) Slope of given lines are m1=12 and m2=–1. Since, AB = AC, then ∠ABC = ∠ACB Now, angle between AB and BC = angle between AC and BC m–121+12(m)=–1–m1–m ⇒ 2m–12+m=m+1m–1 ⇒ 2m2–3m+1=m2+3m+2 ⇒ m2–6m–1=0 Sum of roots = 6 ∴ Required sum is 6.

Want Us to Email you About Special Offers & Updates?

Q.
Two equal sides of an isosceles triangle are along – x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is: [2025]

1 $- 2 \sqrt{10}$