Question 43 of 70intermediate🔧 ApplyShort Answer4 marks

Find the coordinates of the foot of perpendicular from the point (−1,3)(-1, 3) to the line 3x−4y−16=03x - 4y - 16 = 0.

Correct Answer

The coordinates of the foot of perpendicular are (68/25, -49/25).

Exercise: EXERCISE 9.3 | Q: 13 | (Page 18)
For More Understanding

Explanation

This problem requires finding the intersection point of the given line with a perpendicular line passing through the given point. The solution uses the concept that the product of slopes of perpendicular lines equals -1. The cross-multiplication method (shown in the context for solving simultaneous linear equations) provides an efficient way to find the intersection point.

Solution Steps

  1. Step 1: Find the slope of given line 3x−4y−16=03x - 4y - 16 = 0. Rewriting: y=34x−4y = \frac{3}{4}x - 4. So slope m1=34m_1 = \frac{3}{4}.

  2. Step 2: Find slope of perpendicular line. Since perpendicular lines have slopes satisfying m1×m2=−1m_1 \times m_2 = -1, we get m2=−43m_2 = -\frac{4}{3}.

  3. Step 3: Write equation of perpendicular line through point (−1,3)(-1, 3) using point-slope form: y−3=−43(x+1)y - 3 = -\frac{4}{3}(x + 1). Simplifying: 4x+3y−5=04x + 3y - 5 = 0.

  4. Step 4: Find intersection point by solving: 3x−4y−16=03x - 4y - 16 = 0 and 4x+3y−5=04x + 3y - 5 = 0. Using cross-multiplication: x(−4)(−5)−(3)(−16)=y(−16)(4)−(−5)(3)=1(3)(3)−(−4)(4)\frac{x}{(-4)(-5) - (3)(-16)} = \frac{y}{(-16)(4) - (-5)(3)} = \frac{1}{(3)(3) - (-4)(4)}.

  5. Step 5: Simplifying: x68=y−49=125\frac{x}{68} = \frac{y}{-49} = \frac{1}{25}. Therefore, x=6825x = \frac{68}{25} and y=−4925y = -\frac{49}{25}.

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