**How to find equation of tangent to the curve without derivative :**

Here we are going to see how to find equation of tangent to the curve without derivative

To find the equation of the tangent, we need to do the following changes in the given equation of curve.

Replace

x^{2} as xx_{1,}

_{y}^{2} as yy_{1, }

x as (x + x_{1})/2

y as (y + y_{1})/2

xy as (xy_{1} + yx_{1})/2

Now we have to apply the given point instead of (x_{1}, y_{1}) in the equation.

Let us look into some examples to understand the above concept.

**Example 1 :**

Find the equation of the tangent to the parabola y^{2} = 12x at the point (3, 6)

**Solution :**

**In the first step, we are going to replace **y^{2} = yy_{1} and x = (x + x_{1})/2

**Equation of tangent at (x _{1}, y_{1})**

**yy _{1} = 12(x + x_{1})/2**

**y(6) = 6(x + 3)**

**6y = 6x + 18**

**6x - 6y + 18 = 0**

**Dividing the equation by 6, we get**

**x - y + 3 = 0**

Hence equation of tangent is **x - y + 3 = 0**

**Example 2 :**

Find the equation of the tangent to the parabola x^{2} + x − 2y + 2 = 0 at (1, 2)

**Solution :**

**In the first step, we are going to replace x**^{2} = xx_{1} and x = (x + x_{1})/2 and y = (y + y_{1})/2

**Equation of tangent at (x _{1}, y_{1})**

xx1 + ((x + x1)/2) − 2(y + y1)/2 + 2 = 0

x(1) + ((x + 1)/2) − (y + 2) + 2 = 0

2x + x + 1 - 2(y + 2) + 4 = 0

3x - 2y - 4 + 1 + 4 = 0

3x - 2y + 1 = 0

Hence equation of tangent is 3x - 2y + 1 = 0

**Example 3 :**

Find the equation of the tangent to the hyperbola 9x^{2}- 5y^{2} = 31 at (2, -1)

**Solution :**

**In the first step, we are going to replace x**^{2} = xx_{1} and y^{2} = yy_{1}

**Equation of tangent at (x _{1}, y_{1})**

9xx_{1}- 5yy_{1 }= 31

Equation of tangent at (2, -1)

9x(2) - 5y(-1)_{ }= 31

18x + 5y = 31

Hence equation of tangent is 18x + 5y = 31

After having gone through the stuff given above, we hope that the students would have understood "How to find equation of tangent to the curve without derivative"

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